Bayesian Monte Carlo Analysis

Environmental data from generically similar sites 

Multisite model site-specific model input data 

Use acceptance-rejection procedure (e.g., Bayes rule) to evaluate predictions consistence with the site-specific model output data 

Application of Uncertainty Analysis to Ecological Risk of Pesticides 

Hornberger and Spear s original application of generalized sensitivity analysis (GSA) used a binary acceptance-rejection procedure, i.e., they discarded a Monte Carlo realization if they thought that the prediction was inconsistent with the site-specific data (a nonbehavior ) or kept it if they thought it was consistent (a behavior ). The prior probability on each Monte Carlo realization was the reciprocal of the total number of realizations. After the acceptance-rejection procedure was applied, the updated (posterior) probability on each realization that was classified as a behavior was the reciprocal of the number of behaviors, and the posterior probability on nonbehaviors was zero. 

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Dakins ME, Toll JE, Small MJ, Brand K. 1996. Risk-based environmental remediation Bayesian Monte Carlo analysis and the expected value of sample information. Risk Anal 16 67-69. 

A Monte Carlo, Bayesian Monte Carlo, and First-Order Error Analysis... 

In conclusion, we believe that error propagation methods like Monte Carlo, Bayesian Monte Carlo, and Ist-order error analysis should be promoted and extensively used in pesticide risk assessments implemented in both the United States and Europe. 

Dilks DW, Canale RP, Meier PG. 1989. Analysis of model uncertainty using Bayesian Monte Carlo. In Proceedings of ASCE Specialty Conference on Environmental Engineering. New York American Society of Civil Engineers, p 571-577. 

Warren-Hicks WJ, Butcher B. 1996. Monte Carlo analysis classical and Bayesian applications. Human Ecol Risk Assess 2 643-649. 

Determine whether there are more cost-effective alternatives to additional data generation and risk assessment refinements. What-if analyses can be used to examine the savings in risk management that might result from additional data generation. Techniques that may be suitable for this include Bayesian Monte Carlo and expected value of information (EVOI) analysis (Dakins et al. 1996). 

For some aspects of model uncertainties, well-known quantification methods are available. A Bayesian approach might be practicable, for instance, to quantify the uncertainty on the probability model to apply for the stochastic failure behaviour of system components. Monte Carlo analysis might be appropriate to quantify the uncertainty resulting from the application of thermal-hydraulics codes. 

In the next subsection, I describe how the basic elements of Bayesian analysis are formulated mathematically. I also describe the methods for deriving posterior distributions from the model, either in terms of conjugate prior likelihood forms or in terms of simulation using Markov chain Monte Carlo (MCMC) methods. The utility of Bayesian methods has expanded greatly in recent years because of the development of MCMC methods and fast computers. I also describe the basics of hierarchical and mixture models. 

Bayesian methods are very amenable to applying diverse types of information. An example provided during the workshop involved Monte Carlo predictions of pesticide disappearance from a water body based on laboratory-derived rate constants. Field data for a particular time after application was used to adjust or update the priors of the Monte Carlo simulation results for that day. The field data and laboratory data were included in the analysis to produce a posterior estimate of predicted concentrations through time. Bayesian methods also allow subjective weight of evidence and objective evidence to be combined in producing an informed statement of risk. 

More recently Brochot et al. [89] reported an extension of the isobolographic approach to interaction studies for convulsant interaction among pelloxacin, norfloxacin, and theophylline in rats. Their contribution is unique in that they started out by explaining pharmacodynamic interactions for two drugs, but then extended the approach to derive an isobol for three drug interaction. In addition they included Bayesian analysis and developed a population model with Markov chain Monte Carlo methods. 

Such peak shapes correspond to special cases of P( ) size distributions, sometimes showing unphysical negative proportions of crystallites at some n values. Clearly, the maximum of information (the size distribution and the area-weighted and volume-weighted average sizes) will be obtained by using the Fourier analysis with a stabilization scheme, or Monte Carlo/Bayesian/maximum entropy methods. ... 

Greenland, S. (2001). Sensitivity analysis, Monte Carlo risk analysis, and Bayesian uncertainty assessment. Risk Anal 21, 579-583. 

Thus, we take advantage of the accuracy, robustness and efficiency of the direct problem solution, to tackle the associated inverse heat transfer problem analysis [26, 27] towards the simultaneous estimation of momentum and thermal accommodation coefficients in micro-channel flows with velocity slip and temperature jump. A Bayesian inference approach is adopted in the solution of the identification problem, based on the Monte Carlo Markov Chain method (MCMC) and the Metropolis-Hastings algorithm [28-30]. Only simulated temperature measurements at the external faces of the channel walls, obtained for instance via infrared thermography [30], are used in the inverse analysis in order to demonstrate the capabilities of the proposed approach. A sensitivity analysis allows for the inspection of the identification problem behavior when the external wall Biot number is also included among the parameters to be estimated. 

Larget, B. and Simon, D. L. (1999) Markov chain Monte Carlo algorithms for the Bayesian analysis of phylogenetic trees. Molecular Biology and Evolution, 16 750-759. 

Inside this framework the uncertainty is described in terms of random variables and their joint probability distribution. Probabilistic dependencies are represented through a Bayesian Network whose nodes correspond to uncertain elements within the modeled physical systems and hazard, as done, e.g., in [2]. However, so far, and contrary to [2] the full power of BNs has not been exploited, and the network is used only in a forward simulation the only probabilistic analysis methods currently employed are, indeed, the plain Monte Carlo... 

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